Dynamics and scaling of one-dimensional surface structures

We study several one-dimensional step flow models. Numerical simulations show that the slope of the profile exhibits scaling in all cases. We apply a scaling ansatz to the various step flow models and investigate their long time evolution. This evolution is described in terms of a continuous step density function, which scales in time according to ${D(x,t)=F(xt}^{\ensuremath{-}1/\ensuremath{\gamma}}).$ The value of the scaling exponent $\ensuremath{\gamma}$ depends on the mass transport mechanism. When steps exchange atoms with a global reservoir the value of $\ensuremath{\gamma}$ is 2. On the other hand, when the steps can only exchange atoms with neighboring terraces, $\ensuremath{\gamma}=4.$ We compute the step density scaling function for three different profiles for both global and local exchange mechanisms. The computed density functions coincide with simulations of the discrete systems. These results are compared to those given by the continuum approach of Mullins.